In this paper, we present a framework for automated shape differentiation in the finite element software NGSolve. Our approach combines the mathematical Lagrangian approach for differentiating PDE-constrained shape functions with the automated differentiation capabilities of NGSolve. The user can decide which degree of automatisation is required, thus allowing for either a more custom-like or black-box–like behaviour of the software. We discuss the automatic generation of first- and second-order shape derivatives for unconstrained model problems as well as for more realistic problems that are constrained by different types of partial differential equations. We consider linear as well as nonlinear problems and also problems which are posed on surfaces. In numerical experiments, we verify the accuracy of the computed derivatives via a Taylor test. Finally, we present first- and second-order shape optimisation algorithms and illustrate them for several numerical optimisation examples ranging from nonlinear elasticity to Maxwell’s equations.

在本文中，我们提出了一个在有​​限元软件NGSolve中自动进行形状微分的框架。我们的方法将用于区分 PDE 约束形状函数的数学拉格朗日方法与NGSolve的自动微分功能相结合. 用户可以决定需要何种程度的自动化，从而允许软件的更像自定义或类似黑盒的行为。我们讨论了无约束模型问题以及受不同类型偏微分方程约束的更现实问题的一阶和二阶形状导数的自动生成。我们考虑线性和非线性问题以及表面上的问题。在数值实验中，我们通过泰勒检验验证了计算导数的准确性。最后，我们介绍了一阶和二阶形状优化算法，并针对从非线性弹性到麦克斯韦方程的几个数值优化示例说明了它们。